Set theory

The theory of sets is a branch of mathematics that studies the general properties of sets - sets of elements of an arbitrary nature that have some common property. Created in the second half of the 19th century by George Cantor with the significant participation of Richard Dedekind , brought into mathematics a new understanding of the nature of infinity , a deep connection of the theory with formal logic was discovered, but already at the end of the 19th and beginning of the 20th centuries the theory encountered significant difficulties in the form of emerging paradoxes, therefore the initial form of the theory is known as the naive set theory . In the XX century, the theory received significant methodological development; several versions of the axiomatic set theory were created , providing a universal mathematical toolkit, in connection with questions of the measurability of sets , a descriptive theory of sets has been carefully developed .

Set theory has become the basis of many branches of mathematics - general topology , general algebra , functional analysis and has had a significant impact on the modern understanding of the subject of mathematics ^[1] . In the first half of the 20th century, the set-theoretical approach was introduced into many traditional branches of mathematics, and therefore began to be widely used in the teaching of mathematics, including in schools. However, the use of set theory for the logically flawless construction of mathematical theories is complicated by the fact that it itself needs to justify its methods of reasoning. Moreover, all the logical difficulties associated with the justification of the mathematical doctrine of infinity, when moving to the point of view of the general theory of sets, acquire only greater severity ^[2] .

Starting from the second half of the 20th century, the idea of the significance of a theory and its influence on the development of mathematics significantly decreased due to the realization of the possibility of obtaining fairly general results in many areas of mathematics and without the explicit use of its apparatus, in particular, using category-theoretic tools (by means of which topos theory generalized almost all variants of set theory). Nevertheless, the notation of set theory has become generally accepted in all branches of mathematics, regardless of the use of the set-theoretic approach. At the end of the 20th century, several generalizations were created on the ideological basis of set theory. , including the theory of fuzzy sets , the theory of multisets (used mainly in applications), the (developed mainly by Czech mathematicians).

Key concepts of the theory : a set (a collection of objects of arbitrary nature), the relation of elements to sets, a subset , operations on sets , a mapping of sets , one-to-one correspondence , power ( finite , countable , uncountable ), transfinite induction .

One of the visualizations of the three-dimensional version of the Cantor set - nowhere dense perfect set

History

Background

Sets, including infinite ones, have been implicitly represented in mathematics since the time of Ancient Greece : for example, relations of inclusion of the sets of all rational, integer, natural, odd, prime numbers were considered in one form or another. The rudiments of the idea of the equal power of sets are found in Galileo : while discussing the correspondence between numbers and their squares , he draws attention to the inapplicability of the axiom "the whole is larger than the part" to infinite objects ( Galileo's paradox ) ^[3] .

The first idea of a truly infinite set is attributed to the works of Gauss in the early 1800s published in his “ Arithmetic Studies ” ^[4] , in which, introducing comparisons on the set of rational numbers, he discovers equivalence classes ( residue classes ) and breaks the whole set into these classes, noting their infinity and mutual correspondence, consider an infinite number of solutions $ax+b\equiv 0{\pmod {n}}$ ${\ displaystyle ax + b \ equiv 0 {\ pmod {n}}}$ as a single set, classifies binary quadratic forms ( $ax^{2}+2bxy+cy^{2}$ ${\ displaystyle ax ^ {2} + 2bxy + cy ^ {2}}$ ), depending on the determinant, and considers this infinite set of classes as infinite collections of objects of non-numeric nature, suggests the possibility of choosing from the equivalence classes one object representing the entire class ^[5] : it uses methods characteristic of the set-theoretic approach, not used explicitly in mathematics until the 19th century. In later works, Gauss, considering the set of complex numbers with rational real and imaginary parts, speaks of real, positive, negative, purely imaginary integers as its subsets ^[6] . However, the infinite sets or classes as independent objects of research by Gauss did not clearly stand out, moreover, Gauss contains statements against the possibility of using actual infinity in mathematical proofs ^[7] .

A more distinct idea of infinite sets is manifested in Dirichlet’s works, in the course of lectures of 1856-1857 ^[8] , built on the basis of the Gaussian “Arithmetic Studies”. The works of Galois , Scheman and Serre on the theory of functional comparisons of the 1820-1850s also outline elements of the set-theoretic approach, which Dedekind generalized in 1857, which clearly formulated as one of the conclusions the need to consider the whole system of infinitely many comparable numbers as a single object , whose general properties are equally inherent in all its elements, and the system of infinitely many incomparable classes likens a series of integers ^[9] . Separate concepts of set theory can be found in the works of Steiner and Staudt of the 1830-1860s on projective geometry : almost the entire subject largely depends on the idea of a one-to-one correspondence , which is key for set theory, however, in projective geometry additional correspondences were superimposed restrictions (preservation of some geometric relationships ). In particular, Steiner explicitly introduces the concept of an uncountable set for a set of points on a straight line and a set of rays in a beam and operates with their uncountable subsets, and in 1867 he introduces the concept of power as characteristics of sets between which it is possible to establish a projective correspondence (Kantor later pointed out that borrowed the concept and term from Steiner, generalizing projective correspondence to one-to-one) ^[10] .

The representations closest to Cantor’s naive set theory are contained in the writings of Bolzano ^[11] , first of all, in the work , published after the author’s death in 1851 , in which arbitrary numerical sets are considered, and their comparison is clearly defined the concept of one-to-one correspondence , and the term “multitude” ( German: menge ) itself was also systematically used for the first time in this work. However, Bolzano's work is more philosophical than mathematical, in particular, there is no clear distinction between the cardinality of the set and the concept of magnitude or order of infinity, and there is no formal or holistic mathematical theory in these representations ^[12] . Finally, the theories of the real number of Weierstrass , Dedekind and Mere created in the late 1850s and published in the early 1860s have much in common with the ideas of the naive set theory in the sense that they consider the continuum as a set formed from rational and irrational points ^{[ 13]} .

Naive Set Theory

Georg Cantor in 1870

Scheme of proof of countability of the set of rational numbers

Schematic idea of the proof of the Cantor-Bernstein theorem

The main creator of the set theory in its naive version is the German mathematician Georg Cantor ; the work of 1870-1872 on the development of the theory of trigonometric series (continuing the work of Riemann ), which introduces the concept of a limit point that is close to modern ^[14] and pushed to create an abstraction of a point set using it, he tries to classify “exceptional sets” (sets of points of divergence of a series, possibly infinite) ^[15] . Having taken an interest in the questions of the equipotentiality of sets, in 1873, Cantor discovers the countability of the set of rational numbers and on the question of the equidistance of the sets of integers and real numbers (the last result was published in 1874 at the insistence of Weierstrass ^[16] ^[17] . In 1877, Cantor proves one-to-one correspondence between $\mathbb {R}$ {\ displaystyle \ mathbb {R}} and $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ (for anyone $n>0$ ${\ displaystyle n> 0}$ ) Cantor shares his first results in correspondence with Dedekind and Weierstrass, who respond with supportive criticism and remarks on evidence, and from 1879 until 1884 published six articles in Mathematische Annalen with the results of studies of infinite point sets ^[18] ^[19] .

In 1877, Dedekind published an article “On the Number of Classes of Finite-Field Ideals”, in which he explicitly symbolically operates with sets — fields , modules , ideals , rings , and uses the inclusion relation for them (using the signs “<” and “>”) , the operations of union (with the “+” sign) and intersection (with the infix “-”), and, in addition, actually comes to the algebra of sets, indicating the duality of the operations of union and intersection, in the Dedekind notation:

(A+B)-(A+C)=A+(B-(A+C))

{\ displaystyle (A + B) - (A + C) = A + (B- (A + C))}

,

(A-B)+(A-C)=A-(B+(A-C))

{\ displaystyle (AB) + (AC) = A- (B + (AC))}

,

in his subsequent works, repeatedly using this result ^[20] . In a 1878 publication on the equipotentiality of continua of a different number of dimensions, Cantor uses set-theoretic operations, citing Dedekind's work. In addition, the concept of cardinality of the set was first explicitly introduced in the same work, the countability of any infinite subset of a countable set was proved, and finite fields of algebraic numbers were proposed as examples of countable sets. Cantor’s result on the equipotentiality of continua of a different number of dimensions attracted widespread attention of mathematicians, and already in the same year several works followed ( , , Net ) with unsuccessful attempts to prove the impossibility of simultaneous continuity and mutual uniqueness of mapping continua of various dimensions ^{[ 21]} ( Brower gave exact proof of this fact in 1911).

In 1880, Cantor formulated two key ideas of set theory - the concept of an empty set and the transfinite induction method. Beginning in 1881, other mathematicians began to use Cantor's methods: Volterra , Dubois-Reymond , , Harnack , mainly in connection with questions about the integrability of functions ^[22] . In 1883, Kantor gives historically the first formal definition of the continuum using the concepts of a perfect set and density of a set (different from modern ones used in the general topology , but fundamentally similar to them), and he also constructs a classic example of a nowhere dense perfect set (known as a Cantor set ) ^[23] , and also explicitly formulates the continuum hypothesis (the assumption of the absence of intermediate powers between the countable set and the continuum, its unprovability in ZFC tags shown by Cohen in 1963 ).

From 1885-1895, the work on the creation of a naive theory of sets was developed primarily in the works of Dedekind (during these 10 years, Cantor publishes only one small work because of illness). So, in the book “What are numbers and what do they serve for?” ^[24] (where the axiomatization of arithmetic, also known as Peano arithmetic, was also constructed for the first time), the results of set theory in greatest generality, obtained by that time, for sets of arbitrary nature (not necessarily numerical), an infinite set is defined as one-to-one with a part of itself, the Cantor – Bernstein theorem was first formulated ^[25] , the algebra of sets was stated, and the properties of set-theoretic operations were established ^[26] . Schroeder in 1895 draws attention to the coincidence of set algebra and propositional calculus , thereby establishing a deep connection between mathematical logic and set theory.

In 1895-1897, Cantor published a cycle of two works, which in general completed the creation of the naive theory of sets ^[27] ^[28] .

Since the beginning of the 1880s, first of all, after the publication of ideas on transfinite induction, the set-theoretic approach was sharply rejected by many major mathematicians of that time, the main opponents at that time were Herman Schwartz and, to the greatest extent, Leopold Kronecker , who believed that only natural numbers and what directly reduces to them can be considered as mathematical objects (his phrase is known that “God created natural numbers, and everything else is the work of human hands” ). A serious discussion developed among the theologians and philosophers regarding set theory, which was mainly critical of the ideas about actual infinity and quantitative differences in this concept ^[29] . Nevertheless, by the end of the 1890s, set theory had become generally recognized, largely due to the reports of Hadamard and Hurwitz at the First International Congress of Mathematicians in Zurich ( 1897 ), which showed examples of the successful use of set theory in analysis , as well as widespread application set-theoretical tools already had significant influence in the mathematical community by Hilbert ^[30] .

Paradoxes

The vagueness of the concept of a set in a naive theory, in which the construction of sets was allowed only on the basis of the collection of all objects that have any property, led to the fact that in the period 1895-1925 a significant series of contradictions was discovered, which made serious doubts about the possibility of using set theory as a fundamental tool, the situation became known as the “ crisis of the foundations of mathematics ” ^[31] .

The contradiction, which leads to the consideration of the set of all ordinal numbers, was first discovered by Cantor in 1895 ^[32] , rediscovered and first published by Burali-Forti ( Italian: Cesare Burali-Forti ) in 1897 , and became known as the Burali-Forti paradox ^[33] . In a letter to Dedekind in 1899, Kantor spoke for the first time about the contradictory nature of the universe as the set of all sets, since the set of all its subsets would have to be equally powerful to itself, not satisfying the principle ${\mathfrak {m}}<2^{\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}} <2 ^ {\ mathfrak {m}}}$ ^[34] , later this antinomy became known as the Cantor paradox . In further correspondence, Kantor proposed to consider the actual sets ( Germ. Mengen ), which can be thought of as a single object, and “manifolds” ( vielheiten ) for complex constructions, in one form or another, this idea was reflected in some late axiomatizations and generalizations ^{[35 ]} .

The most significant contradiction that influenced the further development of the theory of sets and foundations of mathematics as a whole was the Russell paradox , discovered around 1901 by Bertrand Russell and published in 1903 in the monograph Foundations of Mathematics . The essence of the paradox is in contradiction when considering the question of belonging to oneself the set of all sets that do not include themselves. In addition, the discovery of antinomies such as the Richard paradox , the Berry paradox and the Grelling-Nelson paradox , showing contradictions in attempts to use self-reference of the properties of elements in constructing sets, dates back to about the same time.

As a result of comprehending the paradoxes that have arisen in the community of mathematicians, two directions have arisen for resolving the problems that have arisen: formalization of set theory by choosing a system of axioms that ensures consistency while maintaining the instrumental power of the theory, and the second is the exclusion from consideration of all constructions and methods that are not intuitively comprehensible. In the framework of the first direction begun by Zermelo , Hilbert , Bernays , Hausdorff , several versions of the axiomatic set theory were created and due to rather artificial limitations, the main contradictions were overcome. The second direction, the main exponent of which was Brower , gave rise to a new direction in mathematics - intuitionism , and to one extent or another it was supported by Poincaré , Lebesgue , Borel , Weil .

Axiomatic set theories

The first axiomatization of set theory in 1908 was published by Zermelo , the central role in eliminating paradoxes in this system was to be played by the “axiom of selection” ( German Aussonderung ), according to which the property $P(x)$ ${\ displaystyle P (x)}$ only then can one form many $\{x\mid P(x)\}$ ${\ displaystyle \ {x \ mid P (x) \}}$ if from $P(x)$ ${\ displaystyle P (x)}$ follows a relationship of the form $x\in A$ ${\ displaystyle x \ in A}$ ^[35] . In 1922, thanks to the work of Skulem and Frenkel, a system based on Zermelo’s axioms was finally formed, including the axioms of volume , the existence of an empty set , pair , sum , degree , infinity, and with and without options . These axiomatics are most widely used and are known as the Zermelo – Frenkel theory , the system with the axiom of choice is denoted by ZFC, without the axiom of choice, by ZF.

The special role of the axiom of choice is associated with its intuitive non-obviousness and the deliberate lack of an effective way to determine the set assembled from the elements of the family. In particular, Borel and Lebesgue believed that the evidence obtained with its use has a different cognitive value than evidence independent of it, while Hilbert and Hausdorff accepted it unconditionally, recognizing it no less obvious than other ZF axioms ^[36] .

Another widely used version of the axiomatization of set theory was developed by von Neumann in 1925 , formalized in the 1930s by Bernays , and simplified by Gödel in 1940 (in the work on proving the independence of the continuum hypothesis from the axiom of choice), the final version became known as a system of axioms von Neumann – Bernays – Gödel and designation NBG ^[37] .

There are a number of other axiomatizations, among them (MK), , .

Descriptive Set Theory

At the beginning of the 20th century, Lebesgue , Baer , and Borel studied the questions of the measurability of sets . Based on these works, the theory of descriptive sets was developed in 1910-1930, which systematically studies the internal properties of sets constructed by set-theoretic operations from objects of relatively simple nature — open and closed sets of Euclidean space , metric spaces , metrizable topological spaces with a countable base . The main contribution to the creation of the theory was made by Luzin , Alexandrov , Suslin , Hausdorf . Since the 1970s, generalizations of the descriptive set theory to the case of more general topological spaces have been developed.

Key Concepts

Venn diagram showing all intersections of the graphemes of the capital letters of the Greek , Russian and Latin alphabets

Cartesian product

\{x,y,z\}\times \{1,2,3\}

{\ displaystyle \ {x, y, z \} \ times \ {1,2,3 \}}

Set theory is based on primary concepts: the set and the membership relation of the set (denoted by $x\in A$ ${\ displaystyle x \ in A}$ ^[38] - " $x$ ${\ displaystyle x}$ there is an element of the set $A$ ${\ displaystyle A}$ "," $x$ ${\ displaystyle x}$ belongs to many $A$ ${\ displaystyle A}$ "). An empty set , usually indicated by $\varnothing$ ${\ displaystyle \ varnothing}$ - a set that does not contain a single element. A subset and a subset are relations of inclusion of one set into another (denoted respectively $A\subseteq B$ ${\ displaystyle A \ subseteq B}$ and $A\supseteq B$ ${\ displaystyle A \ supseteq B}$ for lax inclusion and $A\subset B$ ${\ displaystyle A \ subset B}$ and $A\supset B$ ${\ displaystyle A \ supset B}$ - for strict).

The following operations are defined on sets:

association , denoted as $A\cup B$ ${\ displaystyle A \ cup B}$ - a set containing all elements of $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ ,
difference , denoted as $A\setminus B$ ${\ displaystyle A \ setminus B}$ less often $A-B$ ${\ displaystyle AB}$ - many elements $A$ ${\ displaystyle A}$ not included $B$ ${\ displaystyle B}$ ,
addition , denoted as $\setminus A$ ${\ displaystyle \ setminus A}$ or $-A$ ${\ displaystyle -A}$ - the set of all elements not included in $A$ ${\ displaystyle A}$ (in systems using the universal set ),
intersection , denoted as $A\cap B$ ${\ displaystyle A \ cap B}$ - many of the elements contained in $A$ ${\ displaystyle A}$ so in $B$ ${\ displaystyle B}$ ,
symmetric difference , denoted as $A\bigtriangleup B$ ${\ displaystyle A \ bigtriangleup B}$ less often $A\;\;\!\!{\dot {-}}\;\;\!\!B$ ${\ displaystyle A \; \; \! \! {\ dot {-}} \; \; \! \! B}$ - the set of elements included in only one of the sets - $A$ ${\ displaystyle A}$ or $B$ ${\ displaystyle B}$ .

Union and intersection are also often considered over families of sets, denoted by $\bigcup {\mathfrak {A}}$ ${\ displaystyle \ bigcup {\ mathfrak {A}}}$ and $\bigcap {\mathfrak {A}}$ ${\ displaystyle \ bigcap {\ mathfrak {A}}}$ and constitute, respectively, the union of all sets in the family ${\mathfrak {A}}$ ${\ displaystyle {\ mathfrak {A}}}$ and the intersection of all sets in the family.

The union and intersection are commutative , associative and idempotent . Depending on the choice of the system of axioms and the presence of a complement, the set algebra (with respect to union and intersection) can form a distributive lattice , a complete distributive lattice, a Boolean algebra . Venn diagrams are used to visualize operations on sets.

Cartesian product of sets $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ - the set of all ordered pairs of elements from $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ : $A\times B=\{(x,y)\mid x\in A\land y\in B\}$ ${\ displaystyle A \ times B = \ {(x, y) \ mid x \ in A \ land y \ in B \}}$ . Display $f$ ${\ displaystyle f}$ many $A$ ${\ displaystyle A}$ in many $B$ ${\ displaystyle B}$ set theory is regarded as a binary relation - a subset $A\times B$ ${\ displaystyle A \ times B}$ - with the condition of uniqueness of correspondence of the first element to the second: $(x,y)\in f\Rightarrow \forall z\neq y((x,z)\notin f)$ ${\ displaystyle (x, y) \ in f \ Rightarrow \ forall z \ neq y ((x, z) \ notin f)}$ .

A bulean is the set of all subsets of a given set, denoted by ${\mathcal {P}}(A)$ ${\ displaystyle {\ mathcal {P}} (A)}$ or $2^{A}$ ${\ displaystyle 2 ^ {A}}$ (since it corresponds to the set of mappings from $A$ ${\ displaystyle A}$ at $\mathbf {2} =\{0,1\}$ ${\ displaystyle \ mathbf {2} = \ {0,1 \}}$ )

Мощность множества (кардинальное число) — характеристика количества элементов множества, формально определяется как класс эквивалентности над множествами, между которыми можно установить взаимно-однозначное соответствие, обозначается $|A|$ ${\displaystyle |A|}$ or $\sharp A$ $\sharp A$ . Мощность пустого множества равна нулю, для конечных множеств — целое число, равное количеству элементов. Над кардинальными числами, в том числе характеризующими бесконечные множества, можно установить отношение порядка , мощность счётного множества обозначается $\aleph _{0}$ $\aleph _{0}$ ( алеф — первая буква еврейского алфавита), является наименьшей из мощностей бесконечных множеств, мощность континуума обозначается ${\mathfrak {c}}$ ${\mathfrak {c}}$ or $2^{\aleph _{0}}$ $2^{\aleph _{0}}$ , континуум-гипотеза — предположение о том, что между счётной мощностью и мощностью континуума нет промежуточных мощностей. ^[39]

Представление порядковых чисел до

\omega ^{\omega }

\omega ^{\omega }

Если кардинальное число характеризует класс эквивалентности множеств относительно возможности установить взаимно-однозначное соответствие, то порядковое число (ординал) — характеристика классов эквивалентности вполне упорядоченных множеств относительно биективных соответствий, сохраняющих отношение полного порядка. Строятся ординалы посредством введения (с операциями сложения и умножения), порядковое число конечных множеств совпадает с кардиналом (обозначается соответствующим натуральным числом), порядковое число множества всех натуральных чисел с естественным порядком обозначается как $\omega$ ${\ displaystyle \ omega}$ , далее конструируются числа:

\omega +1,\omega +2,\dots ,\omega \cdot 2,\omega \cdot 2+1,\dots ,\omega ^{2},\dots \omega ^{\omega },\dots ,\omega ^{\omega ^{\omega }},\dots ,

\omega +1,\omega +2,\dots ,\omega \cdot 2,\omega \cdot 2+1,\dots ,\omega ^{2},\dots \omega ^{\omega },\dots ,\omega ^{\omega ^{\omega }},\dots ,

,

после чего вводятся $\varepsilon _{0}$ -числа :

\varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup\{\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \}

\varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup\{\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \}

.

Мощность множества всех $\omega$ ${\ displaystyle \ omega}$ - и $\varepsilon$ $\varepsilon$ -чисел — счётных ординалов, обладает мощностью $\aleph _{1}$ $\aleph _{1}$ . ^[40]

Обобщения

Средствами теории категорий , зачастую противопоставляемой теории множеств и с инструментальной, и с дидактической точек зрения, Ловер и Тирни ( англ. Miles Tierney ) в 1970 году создали теорию топосов , изучаемый ею объект — элементарный топос — построен по принципу схожести с поведением множеств в теоретико-множественном понимании, элементарными топосами удалось представить практически все варианты теории множеств.

Теория нечётких множеств — расширение теории множеств, предложенное в 1960-х годах Лотфи Заде ^[41] в рамках концепции нечёткой логики , в нечёткой теории вместо отношения принадлежности элементов к множеству рассматривается функция принадлежности со значениями в интервале $[0,1]$ ${\displaystyle [0,1]}$ : элемент чётко не принадлежит множеству если функция его принадлежности равна нулю, чётко принадлежит — если единице, в остальных случаях отношение принадлежности считается нечётким. Применяется в теории информации , кибернетике , информатике .

Теория мультимножеств ^[42] , в применении к теории сетей Петри называемая теорией комплектов, рассматривает в качестве основного понятия наборы элементов произвольной природы, в отличие от множества, допускающие присутствие нескольких экземпляров одного и того же элемента, отношение включения в этой теории заменено функцией числа экземпляров: $\sharp (a,A)$ $\sharp (a,A)$ — целое число вхождений элемента $a$ ${\displaystyle a}$ в мультимножество $A$ ${\displaystyle A}$ , при объединении комплектов число экземпляров элементов берётся по максимуму вхождений ( $\sharp (a,A_{1}\cup A_{2})=\max(\sharp (a,A_{1}),\sharp (a,A_{2})$ $\sharp (a,A_{1}\cup A_{2})=\max(\sharp (a,A_{1}),\sharp (a,A_{2})$ ), при пересечении — по минимуму ( $\sharp (a,A_{1}\cap A_{2})=\min(\sharp (a,A_{1}),\sharp (a,A_{2})$ $\sharp (a,A_{1}\cap A_{2})=\min(\sharp (a,A_{1}),\sharp (a,A_{2})$ ) ^[43] . Используется в теоретической информатике , искусственном интеллекте , теории принятия решений .

— теория, развиваемая чехословацкими математиками с 1970-х годов, в основном в работах Петра Вопенки ( чеш. Petr Vopěnka ) ^[44] , основывающаяся на чёткой формализации множества как объекта, индуктивно построимого из пустого множества и заведомо существующих элементов, для свойств объектов, допускающих рассмотрения их в целой совокупности, вводится понятие классов, а для изучения подклассов множеств используется концепция .

In Culture

«Теоретико-множественные» часы в Берлине показывают время 9:32

В 1960—1970-е годы в рамках теории музыки была создана собственная , предоставляющая средства чрезвычайно обобщённого описания музыкальных объектов ( звуков с их высотами , динамикой , длительностью ), взаимоотношения между ними и операции над их группами (такими как транспозиция , обращение ). Однако связь с математической теорией множеств более чем опосредованная, и, скорее, терминологическая и культурная: в музыкальной теории множеств рассматриваются только конечные объекты и каких-то существенных теоретико-множественных результатов или значительных конструкций не используется; гораздо в большей степени в этой теории задействованы аппараты теории групп и комбинаторики ^[45] .

Также в большей степени под культурным, нежели содержательным влиянием теории множеств немецким дизайнером Биннингером ( нем. Dieter Binninger ) в 1975 году были созданы так называемые «теоретико-множественные» часы ( нем. Mengenlehreuhr ) (также известны как берлинские часы, нем. Berlin-Uhr ), вошедшие в Книгу рекордов Гиннеса как первое устройство, использующее пятеричный принцип для отображения времени посредством цветных светящихся индикаторов (первый и второй ряд индикаторов сверху показывает часы, третий и четвёртый — минуты; каждый светящийся индикатор соответствует пяти часам для первого ряда, одному часу для второго ряда, пяти минутам для третьего ряда и одной минуте для четвёртого ряда). Часы установлены в берлинском торгово-офисном комплексе Europa-Center .

Notes

↑ Множеств теория / П. С. Александров // Большая советская энциклопедия : [в 30 т.] / гл. ed. A.M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia, 1969-1978. « <…>явилась фундаментом ряда новых математических дисциплин (теории функций действительного переменного, общей топологии, общей алгебры, функционального анализа и др.) <…> оказала глубокое влияние на понимание самого предмета математики »
↑ Математический энциклопедический словарь. — М. : «Сов. энциклопедия » , 1988. — С. 382.
↑ Бурбаки, 1963 , с. 39.
↑ CF Gauss . Disquititiones arithmeticae. — Lipsiae , 1801.
↑ Медведев, 1965 , с. 15—17.
↑ Медведев, 1965 , с. 22—23.
↑ Медведев, 1965 , с. 24.
↑ PG Lejuen Dirichlet . Vorlesungen über Zahlentheorie. — Braunschweig, 1863. , курс к изданию готовил Дедекинд , уже после смерти Дирихле
↑ Медведев, 1965 , с. 24—27.
↑ Медведев, 1965 , с. 28—32.
↑ Медведев, 1965 , с. 74—77.
↑ Бурбаки, 1963 , с. 39—40.
↑ Медведев, 1965 , с. 61—67.
↑ Медведев, 1965 , с. 86—87.
↑ Бурбаки, 1963 , с. 40.
↑ Медведев, 1965 , с. 94—95.
↑ Кантор, 1985 , 2. Об одном свойстве совокупности всех алгебраических чисел. Оригинал: Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. — Journal für die Reine und Angewandte Mathematik, 77 (1874), p. 258—262, с. 18—21.
↑ Кантор, 1985 , 5. О бесконечных линейных точечных многообразиях. Оригинал: Über unendliche, lineare Punktmannichfahltigkeiten. — Mathematische Annalen, Bd. 15 (1879), 17 (1880), 20 (1882), 21 (1883), 23 (1884), с. 40—141.
↑ Бурбаки, 1963 , с. 40—41.
↑ Медведев, 1965 , с. 103—105.
↑ Медведев, 1965 , с. 107—110.
↑ Медведев, 1965 , с. 113—117.
↑ Медведев, 1965 , с. 126—131.
↑ Dedekind, R. Was sind und was sollen die Zahlen? . — Braunschweig: Drud und Berlag von Friedrich Bieweg, 1893. — 60 p.
↑ Доказана независимо Эрнстом Шрёдером и Феликсом Бернштейном в 1897 году
↑ Медведев, 1965 , 14. «Что такое числа и для чего они служат?» Р. Дедекинда, с. 144—157.
↑ Кантор, 1985 , 10. К обоснованию учения о трансфинитных множествах. Оригинал: Beiträge zur Begründung der transfiniten Mengenlehre. — Mathematische Annalen, Bd. 46 (1895) p. 481—512; Bd. 49 (1897), p. 207—246, с. 173—245.
↑ Медведев, 1965 , 17. Новый взлёт Кантора, с. 171—178.
↑ Медведев, 1965 , с. 133—137.
↑ Бурбаки, 1963 , «Никто не сможет изгнать нас из рая, созданного для нас Кантором» — говорит Гильберт в «Основаниях геометрии», изданных в 1899 году, с. 44,49.
↑ Бурбаки, 1963 , Парадоксы теории множеств и кризис оснований, с. 44—53.
↑ Не опубликовано, сообщено в письме Гильберту
↑ Медведев, 1965 .
↑ Бурбаки, 1963 , с. 44.
↑ ¹ ² Бурбаки, 1963 , с. 46.
↑ Куратовский, Мостовский, 1970 , с. 61.
↑ Бурбаки, 1963 , с. 46—47.
↑ Символ $\in$ $\in$ (от греч. εστι — «быть») введён Пеано .
↑ Куратовский, Мостовский, 1970 , с. 176—211, 305—327.
↑ Куратовский, Мостовский, 1970 , с. 273—303.
↑ L. Zadeh . Fuzzy Sets (англ.) // Information and Control. — 1965. — Vol. 5 . — P. 338—353 . — ISSN 0019-9958 . — DOI : 10.1016/S0019-9958(65)90241-X . Архивировано 27 ноября 2007 года.
↑ А. Б. Петровский. Пространства множеств и мультимножеств . — М. : Едиториал УРСС, 2003. — С. 248. — ISBN 5-7262-0633-9 .
↑ Джеймс Питерсон. Обзор теории комплектов // Теория сетей Петри и моделирование систем = Petri Net Theory and The Modelling of Systems. — М. : Мир , 1984. — С. 231—235. — 264 с. — 8400 экз.
↑ П. Вопенка. Математика в альтернативной теории множеств = Mathematics in The Alternative Set Theory / перевод А. Драгалина. — М. : Мир, 1983. — 152 с. — (Новое в зарубежной математике). — 6000 экз.
↑ M. Schuijer. Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. — Rochester : University Rochester Press, 2008. — 306 p. — ISBN 978-1-58046-270-9 .

Literature

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Г. Кантор. Труды по теории множеств. — М. : Наука, 1985. — 430 с. — (Классики науки). — 3450 экз. .
П. Дж. Коэн . Об основаниях теории множеств (рус.) = PJ Cohen, Comments on the foundations of set theory, Proc. Sym. Pure Math. 13 :1 (1971), 9–15. // Успехи математических наук / Ю. И. Манин (перевод). — М. , 1974. — Т. XXIX , вып. 5 (179) . — С. 169—176 . — ISSN 0042-1316 .
К. Куратовский , А. Мостовский . Теория множеств / Перевод с английского М. И. Кратко под редакцией А. Д. Тайманова. — М. : Мир, 1970. — 416 с.
F.A. Medvedev . The development of set theory in the 19th century. - M .: Nauka, 1965 .-- 232 p. - 2500 copies.
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[1] Множеств теория / П. С. Александров // Большая советская энциклопедия : [в 30 т.] / гл. ed. A.M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia, 1969-1978. « <…>явилась фундаментом ряда новых математических дисциплин (теории функций действительного переменного, общей топологии, общей алгебры, функционального анализа и др.) <…> оказала глубокое влияние на понимание самого предмета математики »

[2] Математический энциклопедический словарь. — М. : «Сов. энциклопедия » , 1988. — С. 382.

[_68bf594f2775128d-3] Бурбаки, 1963 , с. 39.

[4] CF Gauss . Disquititiones arithmeticae. — Lipsiae , 1801.

[_e35dc5e435c3eee1-5] Медведев, 1965 , с. 15—17.

[_981124b4c3c4f1ae-6] Медведев, 1965 , с. 22—23.

[_785153abbfc1a25d-7] Медведев, 1965 , с. 24.

[8] PG Lejuen Dirichlet . Vorlesungen über Zahlentheorie. — Braunschweig, 1863. , курс к изданию готовил Дедекинд , уже после смерти Дирихле

[_d4a7d15910fc795c-9] Медведев, 1965 , с. 24—27.

[_3fd32fef6b4b7f8e-10] Медведев, 1965 , с. 28—32.

[_ef6df5612be68c66-11] Медведев, 1965 , с. 74—77.

[_e0905c26688cee9d-12] Бурбаки, 1963 , с. 39—40.

[_f5d02ca0b145492f-13] Медведев, 1965 , с. 61—67.

[_d8b214e98aad8b9e-14] Медведев, 1965 , с. 86—87.

[_68bf584f27751131-15] Бурбаки, 1963 , с. 40.

[_020534f2d6f26c08-16] Медведев, 1965 , с. 94—95.

[_63c9217a27454857-17] Кантор, 1985 , 2. Об одном свойстве совокупности всех алгебраических чисел. Оригинал: Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. — Journal für die Reine und Angewandte Mathematik, 77 (1874), p. 258—262, с. 18—21.

[_33be59280a7d0c9b-18] Кантор, 1985 , 5. О бесконечных линейных точечных многообразиях. Оригинал: Über unendliche, lineare Punktmannichfahltigkeiten. — Mathematische Annalen, Bd. 15 (1879), 17 (1880), 20 (1882), 21 (1883), 23 (1884), с. 40—141.

[_5147578f2bea7618-19] Бурбаки, 1963 , с. 40—41.

[_2687ccdaa9af2801-20] Медведев, 1965 , с. 103—105.

[_54ee7be9ca45ebc3-21] Медведев, 1965 , с. 107—110.

[_5ef4690dab42474f-22] Медведев, 1965 , с. 113—117.

[_1188757a321afeff-23] Медведев, 1965 , с. 126—131.

[24] Dedekind, R. Was sind und was sollen die Zahlen? . — Braunschweig: Drud und Berlag von Friedrich Bieweg, 1893. — 60 p.

[25] Доказана независимо Эрнстом Шрёдером и Феликсом Бернштейном в 1897 году

[_ad8623a63ad6ddbb-26] Медведев, 1965 , 14. «Что такое числа и для чего они служат?» Р. Дедекинда, с. 144—157.

[_a1f44af81f2ae1b7-27] Кантор, 1985 , 10. К обоснованию учения о трансфинитных множествах. Оригинал: Beiträge zur Begründung der transfiniten Mengenlehre. — Mathematische Annalen, Bd. 46 (1895) p. 481—512; Bd. 49 (1897), p. 207—246, с. 173—245.

[_cd2fe8de99f62ca8-28] Медведев, 1965 , 17. Новый взлёт Кантора, с. 171—178.

[_c2b9ffeea8519e5f-29] Медведев, 1965 , с. 133—137.

[_18fa27639dd1aaeb-30] Бурбаки, 1963 , «Никто не сможет изгнать нас из рая, созданного для нас Кантором» — говорит Гильберт в «Основаниях геометрии», изданных в 1899 году, с. 44,49.

[_dff9b91e544635e6-31] Бурбаки, 1963 , Парадоксы теории множеств и кризис оснований, с. 44—53.

[32] Не опубликовано, сообщено в письме Гильберту

[_476e325c56e8bb3b-33] Медведев, 1965 .

[_68bf584f27751135-34] Бурбаки, 1963 , с. 44.

[_68bf584f27751137-35] ¹ ² Бурбаки, 1963 , с. 46.

[_f001183f1ef393e6-36] Куратовский, Мостовский, 1970 , с. 61.

[_1b1f3099e4348df8-37] Бурбаки, 1963 , с. 46—47.

[38] Символ $\in$ $\in$ (от греч. εστι — «быть») введён Пеано .

[_fac0edf73807f813-39] Куратовский, Мостовский, 1970 , с. 176—211, 305—327.

[_de29e4454da06dd1-40] Куратовский, Мостовский, 1970 , с. 273—303.

[41] L. Zadeh . Fuzzy Sets (англ.) // Information and Control. — 1965. — Vol. 5 . — P. 338—353 . — ISSN 0019-9958 . — DOI : 10.1016/S0019-9958(65)90241-X . Архивировано 27 ноября 2007 года.

[42] А. Б. Петровский. Пространства множеств и мультимножеств . — М. : Едиториал УРСС, 2003. — С. 248. — ISBN 5-7262-0633-9 .

[43] Джеймс Питерсон. Обзор теории комплектов // Теория сетей Петри и моделирование систем = Petri Net Theory and The Modelling of Systems. — М. : Мир , 1984. — С. 231—235. — 264 с. — 8400 экз.

[44] П. Вопенка. Математика в альтернативной теории множеств = Mathematics in The Alternative Set Theory / перевод А. Драгалина. — М. : Мир, 1983. — 152 с. — (Новое в зарубежной математике). — 6000 экз.

[45] M. Schuijer. Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. — Rochester : University Rochester Press, 2008. — 306 p. — ISBN 978-1-58046-270-9 .